Given that choosing adjacent beams is not allowed, at most one beam in each group can be chosen. How many ways you can choose five beams from different groups of beams?Explain

Description

Applying a Heuristic

Corresponding reading: Chapter 2, Page 4

Purpose: Investigating the impact of applying a heuristic on the computation time.

Consider the radiation therapy treatment planning discussed in Chapter 2. Assume there are 30 coplanar candidate beams. If the physician wants to choose five beams, how many combinations are possible?

Given that the impact of adjacent beams is similar, a good heuristic to reduce the computation time is excluding configurations in which two or more beams of the five selected beams are adjacent. We want to find an upper bound on the number of possible configurations in which no two beams are adjacent.

Group each two adjacent beams into one group . How many groups will there be?

Given that choosing adjacent beams is not allowed, at most one beam in each group can be chosen. How many ways you can choose five beams from different groups of beams?

Comparing the number of possible configurations before and after grouping beams, calculate the percentage reduction in the number of possible configurations, and hence, the computation time.

Explain why the actual number of possible configurations in which no adjacent beams are chosen is indeed smaller than the number calculated above.

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